17 Introduction to Evidence-Based Robust Optimisation
563
For each level q, given a fixed value of the coupling functions, one can study each
g i independently of the others. The idea is to run an optimisation for each function
g i over only the uncoupled vector u i . With the example in Fig. 17.1 in mind, having
ˆ
h
q
ij (u i ) := h ij ( ˜
d, u i , ˆ
u
q
ij )
where ˆ
u
q
ij := ˆ
u
q
k ∗ ,ij : k ∗ = arg max k F (ˆ u
q
k,ij ), is one of the maxima attained by the
coupled variable u ij . For every Focal Element θ k,i ∈ Θ i we have
ˆ
u
q
k 1 ,1 = arg max
u 1 ∈θ k,1
g 1 ( d, u 1 , h
q
12 (u 1 ), h
q
13 (u 1 )), ∀θ k 1 ,1 ⊂ Θ 1
ˆ
u
q
k 2 ,2 = arg max
u 2 ∈θ k,2
g 2 ( d, u 2 , h
q
21 (u 2 ), h
q
23 (u 2 )), ∀θ k 2 ,2 ⊂ Θ 2
ˆ
u
q
k 3 ,3 = arg max
u 3 ∈θ k,3
g 3 ( d, u 3 , h
q
31 (u 3 ), h
q
32 (u 3 )), ∀θ k 3 ,3 ⊂ Θ 3
(17.55)
with the corresponding values ˆ
g
q
k 1 ,1 , ˆ
g
q
k 2 ,2 and ˆ
g
q
k 3 ,3 .
Once all the maxima over the focal elements of the uncoupled variables are
available for each sample q one can calculate an approximation of Bel(F (d, u) <
ν) as follows. From Eq. (17.55), for each sample q the maximum associated to the
FE θ k = θ k 1 ,1 ×θ k 2 ,2 ×θ k 3 ,3 , for k = 1, . . . N F E,1 ·N F E,2 ·N F E,3 , given the condition
of positive semidefinition of g i , is
max
(u 1 ,u 2 ,u 3 )∈θ k
F ( ˜
d, u 1 , u 2 , u 3 , ˆ
u
q
12 , ˆ
u
q
13 , ˆ
u
q
23 ) = ˆ
g
q
k 1 ,1 + ˆ
g
q
k 2 ,2 + ˆ
g
q
k 3 ,3
(17.56)
with associated basic probability assignment:
bpa
q (θ k ) = bpa(θ k 1 ,1 )bpa(θ k 2 ,2 )bpa(θ k 3 ,3 )ΔBel
q
(17.57)
where ΔBel q =
ij ΔBel
q
ij are the contributions of the partial belief curves
in (17.54): the generic ΔBel
q
ij is the difference of belief between sample q and
q − 1 in the partial belief curve Bel c about the coupled uncertain vector u ij . In other
words, the bpa of each θ k is the product of all the bpa’s of the FE of each uncoupled
variable scaled with the product of the belief values of the samples drawn from the
partial belief curves (Line 18). The approximation of the belief is then computed as:
Bel(F (d, u) ≤ ν) =
q
k
bpa
q (θ k )
(17.58)
563
For each level q, given a fixed value of the coupling functions, one can study each
g i independently of the others. The idea is to run an optimisation for each function
g i over only the uncoupled vector u i . With the example in Fig. 17.1 in mind, having
ˆ
h
q
ij (u i ) := h ij ( ˜
d, u i , ˆ
u
q
ij )
where ˆ
u
q
ij := ˆ
u
q
k ∗ ,ij : k ∗ = arg max k F (ˆ u
q
k,ij ), is one of the maxima attained by the
coupled variable u ij . For every Focal Element θ k,i ∈ Θ i we have
ˆ
u
q
k 1 ,1 = arg max
u 1 ∈θ k,1
g 1 ( d, u 1 , h
q
12 (u 1 ), h
q
13 (u 1 )), ∀θ k 1 ,1 ⊂ Θ 1
ˆ
u
q
k 2 ,2 = arg max
u 2 ∈θ k,2
g 2 ( d, u 2 , h
q
21 (u 2 ), h
q
23 (u 2 )), ∀θ k 2 ,2 ⊂ Θ 2
ˆ
u
q
k 3 ,3 = arg max
u 3 ∈θ k,3
g 3 ( d, u 3 , h
q
31 (u 3 ), h
q
32 (u 3 )), ∀θ k 3 ,3 ⊂ Θ 3
(17.55)
with the corresponding values ˆ
g
q
k 1 ,1 , ˆ
g
q
k 2 ,2 and ˆ
g
q
k 3 ,3 .
Once all the maxima over the focal elements of the uncoupled variables are
available for each sample q one can calculate an approximation of Bel(F (d, u) <
ν) as follows. From Eq. (17.55), for each sample q the maximum associated to the
FE θ k = θ k 1 ,1 ×θ k 2 ,2 ×θ k 3 ,3 , for k = 1, . . . N F E,1 ·N F E,2 ·N F E,3 , given the condition
of positive semidefinition of g i , is
max
(u 1 ,u 2 ,u 3 )∈θ k
F ( ˜
d, u 1 , u 2 , u 3 , ˆ
u
q
12 , ˆ
u
q
13 , ˆ
u
q
23 ) = ˆ
g
q
k 1 ,1 + ˆ
g
q
k 2 ,2 + ˆ
g
q
k 3 ,3
(17.56)
with associated basic probability assignment:
bpa
q (θ k ) = bpa(θ k 1 ,1 )bpa(θ k 2 ,2 )bpa(θ k 3 ,3 )ΔBel
q
(17.57)
where ΔBel q =
ij ΔBel
q
ij are the contributions of the partial belief curves
in (17.54): the generic ΔBel
q
ij is the difference of belief between sample q and
q − 1 in the partial belief curve Bel c about the coupled uncertain vector u ij . In other
words, the bpa of each θ k is the product of all the bpa’s of the FE of each uncoupled
variable scaled with the product of the belief values of the samples drawn from the
partial belief curves (Line 18). The approximation of the belief is then computed as:
Bel(F (d, u) ≤ ν) =
q
k
bpa
q (θ k )
(17.58)